On uniform measures in the Heisenberg group

We initiate a classification of uniform measures in the first Heisenberg group ℍ equipped with the Korányi metric dH, that represents the first example of a noncommutative stratified group equipped with a homogeneous distance. We prove that 1-uniform measures are proportional to the spherical 1-Hausdorff measure restricted to an affine horizontal line, while 2-uniform measures are proportional to spherical 2-Hausdorff measure restricted to an affine vertical line. It remains an open question whether 3-uniform measures are proportional to the restriction of spherical 3-Hausdorff measure to an affine vertical plane. We establish this conclusion in case the support of the measure is a vertically ruled surface. Along the way, we derive asymptotic formulas for the measures of small extrinsic balls in (ℍ,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in ℍ.